A033844 -id:A033844 - cp's OEIS frontend

A340691 Greatest image of A001222 over the prime indices of n.

0, 0, 1, 0, 1, 1, 2, 0, 1, 1, 1, 1, 2, 2, 1, 0, 1, 1, 3, 1, 2, 1, 2, 1, 1, 2, 1, 2, 2, 1, 1, 0, 1, 1, 2, 1, 3, 3, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 2, 1, 1, 2, 4, 1, 1, 2, 3, 2, 1, 1, 3, 1, 2, 0, 2, 1, 1, 1, 2, 2, 3, 1, 2, 3, 1, 3, 2, 2, 2, 1, 1, 1, 1, 2, 1, 2, 2, 1

Offset: 1

Gus Wiseman, Jan 28 2021

nonn

For the initial term, we assume the empty set has maximum image 0.

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The prime indices of 4070 are {1,3,5,12} -> {0,1,1,3}, so a(4070) = 3.
The prime indices of 8892 are {1,1,2,2,6,8} -> {0,0,1,1,2,3} so a(8892) = 3.

Positions of first appearances are A033844.
Positions of 0's are A000079.
Positions of terms <= 1 are A302540.
Positions of 1's are A302540 \ A000079.
The version for minimum is A340928.
A003963 multiplies together the prime indices.
A056239 adds up the prime indices.
A061395 selects the greatest prime index.
A072233 counts partitions by sum and maximum.
A112798 lists the prime indices of each positive integer.
A303975 counts distinct prime factors in the product of prime indices.
Cf. A001222, A006530, A062447, A106529, A244990, A244991, A324522, A340606, A340609, A340610, A340856.

Table[If[n==1,0,Max@@PrimeOmega/@PrimePi/@First/@FactorInteger[n]],{n,100}]

A055739 [e^n]-th prime.

Original entry on oeis.org

2, 3, 17, 71, 251, 857, 2767, 8803, 27211, 82939, 249779, 744949, 2201273, 6463081, 18858529, 54764947, 158330573, 456016933, 1309050653, 3746543923, 10694444393, 30453898201, 86534078387, 245401348403, 694683409429, 1963275871663, 5540095680547, 15611517864749

Offset: 0

Robert G. Wilson v, Jun 09 2000

Amiram Eldar, Table of n, a(n) for n = 0..39

Cf. A000149, A033844, A006988.

Table[Prime[Floor[N[E^n]]],{n,0,25}] (* Typographical error corrected by Harvey P. Dale, Dec 27 2019 *)

a(n) = prime(A000149(n)). - Amiram Eldar, Jul 22 2025

a(25)-a(27) from Amiram Eldar, Jul 22 2025

A065857 The (10^n)-th composite number.

Original entry on oeis.org

4, 18, 133, 1197, 11374, 110487, 1084605, 10708555, 106091745, 1053422339, 10475688327, 104287176419, 1039019056246, 10358018863853, 103307491450820, 1030734020030318, 10287026204717358, 102692313540015924, 1025351434864118026, 10239531292310798956, 102270102190290407386

Offset: 0

Labos Elemer, Nov 26 2001

			The 100th composite number is C(100)=133, while the 100th prime is 541. In general: A000720(m) < A062298(m) < m < A002808(m) < A000040(m), for example pi(100)=25 < 75 < 100 < C(100)=133 < prime(100)=541.

A. E. Bojarincev, Asymptotic expressions for the n-th composite number. Univ. Mat. Zap. 6:21-43(1967). [in Russian]
J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 133, p. 45, Ellipses, Paris 2008.

Chai Wah Wu, Table of n, a(n) for n = 0..22
Laurentiu Panaitopol, Some properties of the series of composed [sic] numbers, Journal of Inequalities in Pure and Applied Mathematics 2:3 (2001).
J. Barkley Rosser and Lowell Schoenfeld, Approximate formulas for some functions of prime numbers. Illinois J. Math. 6 (1962), pp. 64-94.

Cf. A033844, A002808, A062298, A000720, A006988, A065855, A065856.

Composite[n_Integer] := Block[ {k = n + PrimePi[n] + 1 }, While[ k != n + PrimePi[k] + 1, k = n + PrimePi[k] + 1]; Return[k]];
Table[Composite[10^n], {n, 0, 9}]

a(n)=my(k=10^n);forcomposite(n=4,2*k+2,if(k--==0,return(n))) \\ Charles R Greathouse IV, May 30 2013

a(n) = A002808(A011557(n)).

a(n) = 10^(n + n/log n + 2n/log^2 + 4n/log^3 n + O(n/log^4 n)). See Bojarincev for an asymptotic expansion. - Charles R Greathouse IV, May 30 2013

More terms from Robert G. Wilson v, Nov 26 2001
a(14) from Lekraj Beedassy, Jul 14 2008
a(15)-a(19) from Chai Wah Wu, Apr 16 2018
a(20) from Chai Wah Wu, Aug 23 2018

A073799 Numbers that begin a run of consecutive integers k such that PrimePi(k) divides 2^k.

Original entry on oeis.org

2, 7, 19, 53, 131, 311, 719, 1619, 3671, 8161, 17863, 38873, 84017, 180503, 386093, 821641, 1742537, 3681131, 7754077, 16290047, 34136029, 71378569, 148948139, 310248241, 645155197, 1339484197, 2777105129, 5750079047, 11891268401, 24563311309, 50685770167, 104484802057, 215187847711

Offset: 1

Labos Elemer, Aug 12 2002

nonn

It seems that each term is a bit larger than twice the previous one.
Runs have lengths 3, 4, 4, 6, 6, 2, 8, 2, 2, 6, 18, 18, 30, 8, 24, 6, 2, 18, ..., respectively.
From Chai Wah Wu, Jan 27 2020: (Start)
Theorem: a(1) = 2 and a(n) = A033844(n) for n > 1. For n > 1, the length of the n-th run is prime(2^n+1)-prime(2^n) = A051439(n)-A033844(n) = A074325(n).
Proof: Let r > 1. If p = prime(2^r), then primepi(p) = 2^r.
primepi(p-1) = 2^r - 1. Since r > 1, 2^r - 1 > 2 and odd and thus does not divide any power of 2.
In addition 2^r < p and thus divides 2^p. This means that p is a term. Let q be such that p < q < prime(2^r+1). Then primepi(q) = 2^r and divides 2^q. Since primepi(q-1) = 2^r and divides 2^(q-1), this means that q does not start a run and thus is not a term.
Let w be such that prime(2^r+1) <= w < prime(2^(r+1)). Then 2^r + 1 <= primepi(w) < 2^(r+1) and does not divide any power of 2. This means that w is not a term.
(End)

Chai Wah Wu, Table of n, a(n) for n = 1..57

Cf. A000079, A000720, A015910, A062173, A064367, A073797, A073798.
Cf. A033844. - R. J. Mathar, Sep 23 2008
Cf. A051439, A074325.

aQ[k_] := Divisible[2^k, PrimePi[k]]; s = {}; len = {}; n = 2; While[Length[s] < 10, While[! aQ[n], n++]; n1 = n; While[aQ[n], n++]; If[n > n1, AppendTo[s, n1]; AppendTo[len, n - n1]]; n++]; s (* Amiram Eldar, Dec 11 2018 *)

a(n) = if(n==1, 2, prime(2^n)); \\ Jinyuan Wang, Mar 01 2020

from sympy import prime
def A073799(n):
    return 2 if n == 1 else prime(2**n) # Chai Wah Wu, Jan 27 2020

Solutions to 2^(x-1) mod PrimePi(x-1) > 0 but 2^x mod PrimePi(x) = 0.

a(n) = A033844(n) for n > 1. - Chai Wah Wu, Jan 27 2020

Edited by Jon E. Schoenfield, Dec 10 2018
a(15)-a(18) from Amiram Eldar, Dec 11 2018
a(19)-a(33) from Chai Wah Wu, Jan 27 2020

A119778 (32^n)-th prime.

Original entry on oeis.org

2, 131, 8161, 386093, 16290047, 645155197, 24563311309, 910399916939, 33089240375501, 1184895616861903, 41935796950796653, 1470194760556507397, 51140670371058101123, 1767268299972575740723, 60730194035557507211347, 2076859014052740233944627

Offset: 0

Jim Snow (jsnow(AT)mitre.org), Jun 22 2006

Andrew Booker, The Nth Prime Page.

Subsequence of A033844.

Cf. A119772, A119773, A119777, A119779.

Prime[32^Range[0,10]] (* Harvey P. Dale, Oct 02 2021 *)

a(n)=prime(32^n) \\ Charles R Greathouse IV, Nov 02 2014

a(n) = A033844(5*n). - Amiram Eldar, Jul 18 2025

a(8)-a(11) from Charles R Greathouse IV, Nov 02 2014

a(12)-a(15) from Amiram Eldar, Jul 18 2025

A119779 64^n-th prime.

Original entry on oeis.org

2, 311, 38873, 3681131, 310248241, 24563311309, 1870358526653, 138666449011757, 10082409897709157, 722285281729443799, 51140670371058101123, 3586789210729460889317, 249598086801961825095881, 17254990129969542495182251

Offset: 0

Jim Snow (jsnow(AT)mitre.org), Jun 22 2006

Andrew Booker, The Nth Prime Page.

Cf. A033844, A119772, A119773, A119777, A119778.

Prime[64^Range[0,6]] (* Harvey P. Dale, Jul 24 2021 *)

a(n) = prime(64^n); \\ Amiram Eldar, Jul 18 2025

a(n) = A033844(6*n) = A119772(3*n) = A119773(2*n). - Amiram Eldar, Jul 18 2025

a(7)-a(13) from Amiram Eldar, Jul 18 2025

A325091 Heinz numbers of integer partitions of powers of 2.

Original entry on oeis.org

1, 2, 3, 4, 7, 9, 10, 12, 16, 19, 34, 39, 49, 52, 53, 55, 63, 66, 70, 75, 81, 84, 88, 90, 94, 100, 108, 112, 120, 129, 131, 144, 160, 172, 192, 205, 246, 254, 256, 259, 311, 328, 333, 339, 341, 361, 370, 377, 391, 434, 444, 452, 465, 545, 558, 592, 598, 609, 614

Offset: 1

Gus Wiseman, Mar 27 2019

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1) * ... * prime(y_k), so these are numbers whose sum of prime indices is a power of 2. A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

1 is in the sequence because it has prime indices {} with sum 0 = 2^(-infinity).

			The sequence of terms together with their prime indices begins:
   1: {}
   2: {1}
   3: {2}
   4: {1,1}
   7: {4}
   9: {2,2}
  10: {1,3}
  12: {1,1,2}
  16: {1,1,1,1}
  19: {8}
  34: {1,7}
  39: {2,6}
  49: {4,4}
  52: {1,1,6}
  53: {16}
  55: {3,5}
  63: {2,2,4}
  66: {1,2,5}
  70: {1,3,4}
  75: {2,3,3}
  81: {2,2,2,2}

Cf. A000720, A001222, A018819, A033844, A056239, A102378, A112798, A131577, A318400, A325092, A325093.

q:= n-> (t-> t=2^ilog2(t))(add(numtheory[pi](i[1])*i[2], i=ifactors(n)[2])):
select(q, [$1..1000])[];  # Alois P. Heinz, Mar 28 2019

Select[Range[100],#==1||IntegerQ[Log[2,Total[Cases[FactorInteger[#],{p_,k_}:>k*PrimePi[p]]]]]&]

A325092 Heinz numbers of integer partitions of powers of 2 into powers of 2.

Original entry on oeis.org

1, 2, 3, 4, 7, 9, 12, 16, 19, 49, 53, 63, 81, 84, 108, 112, 131, 144, 192, 256, 311, 361, 719, 931, 1197, 1539, 1596, 1619, 2052, 2128, 2401, 2736, 2809, 3087, 3648, 3671, 3969, 4116, 4864, 5103, 5292, 5488, 6561, 6804, 7056, 8161, 8748, 9072, 9408, 11664, 12096

Offset: 1

Gus Wiseman, Mar 27 2019

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1) * ... * prime(y_k), so these are numbers whose prime indices are powers of 2 and whose sum of prime indices is also a power of 2. A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

1 is in the sequence because it has prime indices {} with sum 0 = 2^(-infinity).

			The sequence of terms together with their prime indices begins:
    1: {}
    2: {1}
    3: {2}
    4: {1,1}
    7: {4}
    9: {2,2}
   12: {1,1,2}
   16: {1,1,1,1}
   19: {8}
   49: {4,4}
   53: {16}
   63: {2,2,4}
   81: {2,2,2,2}
   84: {1,1,2,4}
  108: {1,1,2,2,2}
  112: {1,1,1,1,4}
  131: {32}
  144: {1,1,1,1,2,2}
  192: {1,1,1,1,1,1,2}
  256: {1,1,1,1,1,1,1,1}
  311: {64}

Cf. A000720, A001222, A018819, A033844, A056239, A102378, A112798, A318400, A325091, A325093.

q:= n-> andmap(t-> t=2^ilog2(t), (l-> [l[], add(i, i=l)])(
      map(i-> numtheory[pi](i[1])$i[2], ifactors(n)[2]))):
select(q, [$1..15000])[];  # Alois P. Heinz, Mar 28 2019

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
pow2Q[n_]:=IntegerQ[Log[2,n]];
Select[Range[1000],#==1||pow2Q[Total[primeMS[#]]]&&And@@pow2Q/@primeMS[#]&]

A335331 a(n) = prime(k) where k is the n-th 7-smooth number.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 37, 43, 47, 53, 61, 71, 73, 89, 97, 103, 107, 113, 131, 149, 151, 173, 181, 197, 223, 227, 229, 251, 263, 281, 307, 311, 349, 359, 379, 409, 419, 433, 463, 503, 521, 541, 571, 593, 613, 659, 691, 701, 719, 761, 809, 827, 853, 863

Offset: 1

David A. Corneth, Jun 01 2020

nonn

At A110069 we look for numbers of the form n = (d_1 + d_2 + ... + d_k)*prime(d_1*d_2*...*d_k) where d_1 d_2 ... d_k is the decimal expansion of n. As the largest prime that can be among the digits of a base-10 number is 7, the product of digits is 7-smooth. Hence the factor prime(d_1*d_2*...*d_k) is a term from this sequence. As lots of numbers have a product of digits of, say, 210^4, it would help to know prime(210^4) in advance. That's a(5817) of this sequence as 210^4 is the 5817th 7-smooth number. Precomputing such numbers is a computational benefit.

David A. Corneth, Table of n, a(n) for n = 1..8862

Cf. A002473, A006988, A033844, A110069.

A055680 (5^n)-th prime.

Original entry on oeis.org

2, 11, 97, 691, 4637, 28687, 171529, 994837, 5653807, 31643347, 174978379, 958200857, 5205329029, 28089376111, 150722620067, 804829625447, 4279524344539, 22671428650501, 119712643014673, 630280777956749, 3309723954037103, 17339016680833993, 90641632321932691, 472916224088644129, 2463008836130170681

Offset: 0

Robert G. Wilson v, Jun 09 2000

Cf. A033844, A006988.

Mathematica
```
Table[Prime[5^n],{n,0,16}]
```

a(n)=prime(5^n) \\ Charles R Greathouse IV, Nov 02 2014

a(17)-a(24) from Charles R Greathouse IV, Nov 02 2014

cp's OEIS Frontend

A340691 Greatest image of A001222 over the prime indices of n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A055739 [e^n]-th prime.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A065857 The (10^n)-th composite number.

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A073799 Numbers that begin a run of consecutive integers k such that PrimePi(k) divides 2^k.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

Extensions

A119778 (32^n)-th prime.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A119779 64^n-th prime.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A325091 Heinz numbers of integer partitions of powers of 2.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

A325092 Heinz numbers of integer partitions of powers of 2 into powers of 2.

Views